Question

Consider a spherical shell of radius $R$ at temperature $T$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $u=\frac{U}{V} ? T^4$ and pressure $p=\frac{1}{3}\bigg(\frac{U}{V}\bigg)$ If the shell now undergoes an adiabatic expansion, the relation between $T$ and $R$ is

• A. $T ? e^{-R}$
• B. $T ? \frac{1}{R}$
• C. $T ? e^{-3R}$
• D. $T ? \frac{1}{R^3}$

Answer 1

The Correct Option is B

To solve this problem, we need to understand the behavior of black body radiation in terms of thermodynamics and its relation to an ideal gas of photons. Let's break it down step-by-step:

1. First, we are given that the internal energy per unit volume \( u = \frac{U}{V} \propto T^4 \). Since \( u \propto T^4 \), we can write: U = aV T^4, where \( a \) is a constant.
2. The pressure \( p \) is given by p = \frac{1}{3} \left( \frac{U}{V} \right) = \frac{1}{3} aT^4.
3. For adiabatic processes of gases, the relation is given in terms of \( \gamma \) (adiabatic index), where: pV^\gamma = \text{constant}.
4. For a photon gas, the adiabatic index \(\gamma\) is \( \frac{4}{3} \). Thus, the relation becomes: pV^{4/3} = \text{constant}.
5. Substitute the expression for pressure: \frac{1}{3} a T^4 V^{4/3} = \text{constant}.
6. We assume the spherical shell volume V = \frac{4}{3} \pi R^3. Substitute this in: T^4 R^4 = \text{constant}.
7. Solving for temperature, we get: T^4 = \frac{\text{constant}}{R^4} or T \propto \frac{1}{R}.
8. Thus, the correct relation between temperature \(T\) and radius \(R\) for an adiabatic expansion of a spherical shell is: T \propto \frac{1}{R}.

Therefore, the correct answer is T \propto \frac{1}{R}. Let's rule out the incorrect options:

• T \propto e^{-R}: This would imply a non-physical exponential decrease not matching adiabatic conditions.
• T \propto e^{-3R}: Similar to the above, this implies an unjustified exponential relationship.
• T \propto \frac{1}{R^3}: This contradicts the derived relation from photon gas dynamics.